Global existence and uniqueness of the density-dependent incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients

Abstract

We consider the global well-posedness of the inhomogeneous incompressible Navier-Stokes-Korteweg system with a general capillary term. Based on the maximal regularity property, we obtain the global existence and uniqueness of solutions to the incompressible Navier-Stokes-Korteweg system with variable viscosity and capillary terms. By assuming the initial density $\rho_0$ is close to a positive constant, additionally, the initial velocity $u_0$ and the initial density $\nabla \rho_0$ are small in critical space $\dot B^{-1+d/p}_{p,1}(\mathbb R^{d})$ $(1<p<d).$ This work relies on the maximal regularity property of the heat equation, of the Stokes equation, and of the Lam\'e equation.